More on Equal Fractions

You can create as many equal fractions as you like. Just multiply (or divide) both the numerator and denominator by the same number.

Sometimes you know the denominator for the equal fraction as in the first example below $$\frac23=\frac{?}{12}.$$ In this case you are looking for a fraction equal to $\displaystyle{\frac23}$ that has a denominator of $12$.

Start with the denominator of the first fraction and decide what you need to either multiply by, or divide by, to get the denominator of the second fraction. Then multiply both the numerator and denominator of the first fraction to get an equal fraction with the desired denominator.

So, for the example $\displaystyle{\frac23=\frac{?}{12}}$ you need to multiply $3$ by $4$ to get $12$. Therefore, it is necessary to multiply both the numerator and the denominator by $4$ to get the fraction, with a denominator of $12$, equal to $\displaystyle{\frac23}$.

The second example $\displaystyle{\frac{12}{18}=\frac{2}{?}}$ requires you to find a fraction equal to $\displaystyle{\frac{12}{18}}$ with a numerator of $2$. This time start with the numerator of the first fraction and decide what you need to either multiply by, or divide by, to get the numerator of the second fraction. In this example divide both the numerator and denominator by $6$. Do you understand why?

The result is the fraction $\displaystyle{\frac23}$. The only number that now divides into the numerator and denominator is 1, so we consider $\displaystyle{\frac23}$ to be the simplest fraction equal to $\displaystyle{\frac{12}{18}}$.

\begin{align*} \frac23 &=\frac{?}{12}\\[1ex] &=\frac{2\times {\color{red}4}}{3\times {\color{red}4}}\\[1ex] &=\frac{8}{12} \end{align*} \begin{align*} \frac{12}{18} &=\frac{2}{?}\\[1ex] &=\frac{12\div {\color{red}6}}{18\div {\color{red}6}}\\[1ex] &=\frac{2}{3} \end{align*}

What are the values for the symbols $?$ and ${\color{red}\triangle}$ for each of the examples below?

\begin{align*} \frac56 &=\frac{?}{18}\\[1ex] &=\frac{5\times {\color{red}\triangle}}{6\times {\color{red}\triangle}}\\[1ex] &=\frac{?}{18} \end{align*} \begin{align*} \frac{8}{12} &=\frac{?}{3}\\[1ex] &=\frac{8\div {\color{red}\triangle}}{12\div {\color{red}\triangle}}\\[1ex] &=\frac{?}{3} \end{align*}

$?=$

${\color{red}\triangle}=$

$?=$

${\color{red}\triangle}=$
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What number is required in place of the $\Delta$ to make the pairs of fractions equal?

$\displaystyle{\frac{8}{16}=\frac{\Delta}{4}}$ $\Delta =$

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$\displaystyle{\frac{12}{15}=\frac{\Delta}{5}}$ $\Delta =$

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$\displaystyle{\frac{25}{75}=\frac{\Delta}{15}}$ $\Delta =$

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$\displaystyle{\frac{200}{250}=\frac{\Delta}{25}}$ $\Delta =$

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Write the simplest fraction for each of the following fractions.

$\displaystyle{\frac{6}{12}}$

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$\displaystyle{\frac{75}{100}}$

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$\displaystyle{\frac{4}{82}}$

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$\displaystyle{\frac{9}{27}}$

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